A084138 -id:A084138 - cp's OEIS frontend

A084141 a(n) is the index in sequence A084138 when the value of that sequence is one (1), i.e., there is exactly one case where there are exactly a(n) primes between m and 2m, exclusively, for m > 0.

0, 22, 36, 47, 79, 98, 114, 134, 173, 178, 184, 210, 218, 221, 245, 254, 262, 284, 297, 305, 327, 333, 373, 387, 396, 426, 459, 466, 470, 484, 530, 544, 563, 567, 575, 587, 616, 650, 694, 700, 706, 708, 737, 776, 859, 881, 885, 898, 926, 939, 974, 993, 1002

Offset: 0

Harry J. Smith, May 15 2003

nonn

This calculation relies on the fact that Pi(2*m) - Pi(m) > m/(3*log(m)) for m >= 5. Conjecture: There are an infinite number of terms in this sequence.

			a(3)=47 because the 3rd one in sequence A084138 is its item 47. There is exactly one case where there are exactly 47 primes between m and 2m.

P. Ribenboim, The Little Book of Big Primes. Springer-Verlag, 1991, p. 140.

Eric Weisstein's World of Mathematics, Bertrand's Postulate.

Cf. A060715, A060756, A084138, A084139, A084140, A084142.

A084140 a(n) is the smallest number j such that if x >= j there are at least n primes between x and 2x exclusively.

Original entry on oeis.org

2, 6, 9, 15, 21, 24, 30, 34, 36, 49, 51, 54, 64, 75, 76, 84, 90, 91, 114, 115, 117, 120, 121, 132, 135, 141, 154, 156, 174, 175, 184, 187, 201, 205, 210, 216, 217, 220, 231, 244, 246, 252, 285, 286, 294, 297, 300, 301, 304, 321, 322, 324, 327, 330, 339, 360, 364

Offset: 1

Harry J. Smith, May 15 2003

nonn

For all m >= a(n) there are at least n primes between m and 2m exclusively. This calculation relies on the fact that pi(2m) - pi(m) > m/(3*log(m)) for m >= 5. This is one more than the terms of A084139 with offset changed from 0 to 1.
For n > 5889, pi(2n) - pi(n) > f(2, 2n) - f(3, n) where f(k, x) = x/log x * (1 + 1/log x + k/(log x)^2). This may be useful for checking larger terms. The constant 3 can be improved at the cost of an increase in the constant 5889. - Charles R Greathouse IV, May 02 2012
A168421(n) = nextprime(a(n)), where nextprime(x) is the next prime >= x. - John W. Nicholson, Dec 21 2012
a(1) = ceiling((A104272(1)+1)/2) modifies the only even prime, 2; which has been stated, in Formula, as a(1) = A104272(1); for all others, a(n) = (A104272(n)+1)/2 = ceiling ((A104272(n)+1)/2). - John W. Nicholson, Dec 24 2012
Srinivasan's Lemma (2014): previousprime(a(n)) = p_(k-n) < (p_k)/2, where the n-th Ramanujan Prime R_n is the k-th prime p_k, and with n > 1. Proof: By the minimality of R_n, the interval ((p_k)/2,p_k] contains exactly n primes, so p_(k-n) < (p_k)/2. - Copied and adapted from a comment by Jonathan Sondow in A168421 by John W. Nicholson, Feb 17 2015

			a(11)=51 since there are at least 11 primes between m and 2m for all m >= 51 and this is not true for any m < 51. Although a(100)=720 is not listed, for all m >= 720, there are at least 100 primes between m and 2m.

Paulo Ribenboim, The Little Book of Big Primes, Springer-Verlag, 1991, p. 140.
Paulo Ribenboim, The Little Book of Bigger Primes, Springer-Verlag, 2004, p. 181.

Vladimir Shevelev, Charles R. Greathouse IV, Peter J. C. Moses, On intervals (kn, (k+1)n) containing a prime for all n>1, Journal of Integer Sequences, Vol. 16 (2013), Article 13.7.3. arXiv version, arXiv:1212.2785 [math.NT], 2012.
Anitha Srinivasan, An upper bound for Ramanujan primes, Integers, 19 (2014), #A19.
Eric Weisstein's World of Mathematics, Bertrand's Postulate.

Cf. A108954, A060715, A060756, A084138, A084139, A084141, A084142, A168421, A168425.

nn = 60;
R = Table[0, {nn}]; s = 0;
Do[If[PrimeQ[k], s++]; If[PrimeQ[k/2], s--]; If[s < nn, R[[s + 1]] = k], {k, Prime[3 nn]}];
A104272 = R + 1;
Ceiling[(A104272 + 1)/2] (* Jean-François Alcover, Nov 07 2018, after T. D. Noe in A104272 *)

a(1) = A104272(1); for n >= 2, a(n) = (A104272(n)+1)/2. - Vladimir Shevelev, Dec 07 2012

a(n) = ceiling((A104272(n)+1)/2) for n >= 1. - John W. Nicholson, Dec 24 2012

A084139 a(n) is the largest number for which exactly n primes are bounded between a(n) and 2a(n) exclusively.

Original entry on oeis.org

1, 5, 8, 14, 20, 23, 29, 33, 35, 48, 50, 53, 63, 74, 75, 83, 89, 90, 113, 114, 116, 119, 120, 131, 134, 140, 153, 155, 173, 174, 183, 186, 200, 204, 209, 215, 216, 219, 230, 243, 245, 251, 284, 285, 293, 296, 299, 300, 303, 320, 321, 323, 326, 329, 338, 359, 363

Offset: 0

Harry J. Smith, May 15 2003

nonn

a(n) is the index of last occurrence of n in A060715. This calculation relies on the fact that Pi(2*m)-Pi(m) > m/(3*Log(m)) for m>=5. It can be shown that every integer >= 0 occurs in A060715, so there is no problem in finding the last occurrence.

A168421(n) = nextprime(a(n)), where nextprime(x) is the next prime > x. Note: some a(n) may be prime, therefore nextprime(x) not equal to x. - John W. Nicholson, Oct 11 2013

			a(10) = 50 since ten primes last arise between 50 and 100: 53, 59, 61, 67, 71, 73, 79, 83, 89, 97.

P. Ribenboim, The Little Book of Big Primes. Springer-Verlag, 1991, p. 140.

T. D. Noe, Table of n, a(n) for n = 0..1000
Eric W. Weisstein, MathWorld: Bertrand's Postulate

Cf. A060715, A060756, A084138, A084140, A084141, A084142.

nn = 100; t = Table[0, {nn}]; Do[m = PrimePi[2*n] - PrimePi[n]; If[0 < m <= nn, t[[m]] = n], {n, 15*nn}]; Join[{1}, t] (* T. D. Noe, Dec 31 2012 *)

a(n) = floor((A104272(n)+1)/2) for n >= 1. - John W. Nicholson, Oct 11 2013

a(n) = A084140(n+1) - 1. - John W. Nicholson, Oct 11 2013

A084142 Positive numbers k such that the number of primes between k and 2k is different from the number of primes between m and 2m for every number m != k.

Original entry on oeis.org

1, 120, 216, 300, 531, 714, 804, 999, 1344, 1356, 1395, 1680, 1764, 1770, 1959, 2046, 2121, 2325, 2484, 2511, 2760, 2826, 3150, 3285, 3396, 3744, 4044, 4116, 4146, 4314, 4710, 4839, 5046, 5070, 5136, 5250, 5586, 5970, 6411, 6459, 6501, 6504, 6846, 7275

Offset: 1

Harry J. Smith, May 15 2003

nonn

The number of primes between k and 2*k is unique because no other number m > 0 has the same of primes between m and 2m, exclusively. k is the value of A060756(j) or A084139(j) when A084138(j) = 1. Question: Is this sequence infinitely long?

If k > 1 is a term then A060715(k-1) < A060715(k) < A060715(k+1). Consequently, (2*k-1, 2*k+1) is a twin prime pair, so 3|k. Additionally, it can be shown that k-1..k+3 are all composite numbers. Moreover, if k is even, then k-4..k+4 are all composite numbers. - Jon E. Schoenfield, Oct 08 2023

			120 is a term because there are 22 primes between 120 and 240 and no other number m > 0 has 22 primes between m and 2*m.

P. Ribenboim, The Little Book of Big Primes. Springer-Verlag, 1991, p. 140.

Eric Weisstein's World of Mathematics, Bertrand's Postulate.

Cf. A060715, A060756, A084138, A084139, A084140, A084141.

Name edited by Jon E. Schoenfield, Oct 08 2023

cp's OEIS Frontend

A084141 a(n) is the index in sequence A084138 when the value of that sequence is one (1), i.e., there is exactly one case where there are exactly a(n) primes between m and 2m, exclusively, for m > 0.

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A084140 a(n) is the smallest number j such that if x >= j there are at least n primes between x and 2x exclusively.

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A084139 a(n) is the largest number for which exactly n primes are bounded between a(n) and 2a(n) exclusively.

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A084142 Positive numbers k such that the number of primes between k and 2*k is different from the number of primes between m and 2*m for every number m != k.

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A084142 Positive numbers k such that the number of primes between k and 2k is different from the number of primes between m and 2m for every number m != k.